1.5 Part - 2 Inverse Relations and Inverse Functions

Transcription

1 1.5 Part - 2 Inverse Relations and Inverse Functions What happens when we reverse the coordinates of all the ordered pairs in a relation? We obviously get another relation, but does it have any similarities to the original relation? If the original relation was a function, will this new relation also be a function? Reversing the coordinates in this way is called finding the inverse of a relation. Let s begin our exploration of inverses by looking at an example. Example 1: Inverse Exploration These are ordered pairs: (-1, 1), (1, 1), (2, 4), (3, 9). They represent the algebraic relationship: This relation is a function because y =. Now let s invert each of the ordered pairs from above: (1, -1), (1, 1), (4, ) (, ). This is still a relation: it relates 1 to -1, and 4 to 2, etc. We call this the. Is this relation a function? Explain why or why not. Definition Inverse Relation The ordered pair (a, b) is in the relation if and only if the ordered pair (, ) is in the inverse relation. In our first example, the inverse relation was not a function even though the original relation was; however, an inverse relation can be a function. We then call it (imaginatively enough) an inverse function. The inverse of a relation is a function if the original relation passes the horizontal line test. Why does this make sense? Page 1

2 Horizontal Line Test The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original in at most. Example 2: Applying the Horizontal Line Test Which graphs are graphs of (a) relations that are functions? (b) relations that have inverses that are functions? Page 2

3 A function whose inverse is a function has a graph that passes both the horizontal and vertical line tests (such as the last graph in Example 3 shown below). Such a function is called one-to-one since every y is paired with a unique x and every x is paired with a unique y. Definition: Inverse Function If f is a function with domain D and range R, then the inverse function of f, denoted f 1, is the function with domain R and range D defined by f 1 (b) = a if and only if f(a) = b Example 3: Finding an Inverse Function Algebraically Find an equation for f 1 (x) if f(x) = x x+1. Sometimes you can find an inverse function algebraically, but many times you can t or it is extremely difficult to do so. In these cases we usually rely on how f maps x to y in order to understand f 1. You can still draw the graph without writing the algebraic equation using the following principle: The Inverse Reflection Principle The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y = x. This means that if you reflect f over the line y = x, you will have drawn f 1. Page 3

4 Example 4: Finding an Inverse Function Graphically The graph of a function y = f(x) is shown in the figure. Sketch a graph of the function y = f 1 (x). There s a connection between inverses and function composition. An inverse function undoes the action of the original function. (So, for example, f(x) = x + 1 and g(x) = x 1 are inverses.) This leads to the following rule: The Inverse Composition Rule A Function f is one-to-one with inverse function g if and only if: f(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f. Example 5: Verifying Inverse Functions Show algebraically that f(x) = x and g(x) = x 1 inverse composition rule. are inverse functions, using the Page 4

5 Finding an Inverse Cheat Sheet Given a formula for a function f, proceed as follows to find a formula for f Determine that there is a function f 1 by checking that f is one-to-one. State any restrictions on the domain of f. 2. Switch x and y in the formula y = f(x). 3. Solve for y. 4. State any restrictions on the domain of and range y = f 1 (x) by interchanging x and y in the domain and range inequalities. Page 5